Practice Maths

Rearranging Formulas

Key Ideas

Key Terms

subject
Of a formula is the variable that stands alone on one side.
inverse operations
In reverse order (undo the last operation first).

Making a New Variable the Subject

The process is identical to solving an equation — isolate the target variable using inverse operations.

Hot Tip Work methodically: (1) identify which variable you want, (2) list what operations are applied to it, (3) undo them in reverse order using inverses. Keep both sides balanced at every step.

Worked Example

Question: Make t the subject of v = u + at.

Step 1 — Subtract u from both sides.
v − u = at

Step 2 — Divide both sides by a.
(v − u)⁄a = t

Step 3 — Write with the subject on the left.
t = (v − u)⁄a

What Does "Changing the Subject" Mean?

Rearranging a formula (also called changing the subject) means rewriting it so that a different variable is isolated on one side. For example, the formula v = u + at has v as the subject. We might need to rearrange it to make a the subject (to find acceleration) or u the subject (to find initial speed).

The process is exactly the same as solving an equation — use inverse operations to move everything except the target variable to the other side. The difference is that letters remain on both sides throughout.

Single-Step Rearrangements

Start with simple formulas that need only one inverse operation.

Example 1: Make m the subject of E = mc2. Divide both sides by c2: m = E/c2.

Example 2: Make C the subject of F = (9/5)C + 32. Subtract 32: F − 32 = (9/5)C. Multiply by 5/9: C = (5/9)(F − 32). This is the Celsius to Fahrenheit conversion rearranged.

Multi-Step Rearrangements

More complex formulas require several steps. A useful strategy: work out what sequence of operations was applied to the target variable in the original formula, then undo them in reverse order.

Example: Make a the subject of v = u + at.

Step 1 — subtract u from both sides: v − u = at.

Step 2 — divide both sides by t: a = (v − u)/t.

Example: Make r the subject of A = πr2.

Step 1 — divide by π: A/π = r2.

Step 2 — take the square root of both sides: r = √(A/π). Note: we take the positive root since r is a length.

Formulas Involving Fractions and Roots

When the target variable is in a denominator, multiply both sides by the denominator first to clear the fraction. When the target variable is under a square root, isolate the root and then square both sides.

Example: Make r the subject of T = 2π√(l/g).

Step 1 — divide by 2π: T/(2π) = √(l/g).

Step 2 — square both sides: T2/(4π2) = l/g.

Step 3 — multiply by g: l = gT2/(4π2). (Here l is the subject, not r — this shows the process for a pendulum formula.)

Why Rearranging Matters

Rearranging formulas is a powerful practical skill. Scientists rearrange E = mc2 to find mass when energy is known. Engineers rearrange speed formulas to find time. In exams, you will often be given a formula and asked to "find x" where x is not the subject — rearranging first makes the calculation straightforward.

Key tip: Write every step on a new line and perform the same operation to both sides simultaneously. If you need to square both sides to remove a square root, remember to square the entire expression on both sides — not just one term. Neat, step-by-step working makes your logic clear and earns method marks even if you make an arithmetic slip.

Mastery Practice

  1. Rearrange each formula to make the indicated variable the subject. Show every step. Fluency

    1. A = lw, make l the subject.
    2. P = 2(l + w), make l the subject.
    3. v = u + at, make u the subject.
    4. v = u + at, make a the subject.
    5. d = st, make s the subject.
    6. d = st, make t the subject.
    7. I = Prt ÷ 100, make r the subject.
    8. I = Prt ÷ 100, make t the subject.
    9. F = ma, make m the subject.
    10. C = 2πr, make r the subject.
    11. y = mx + c, make x the subject.
    12. y = mx + c, make m the subject.

  2. Rearrange each formula to make the indicated variable the subject. These involve squares or square roots. Fluency

    1. A = πr², make r the subject.
    2. E = ½mv², make v the subject.
    3. c² = a² + b², make a the subject.
    4. V = ¾πr³, make r the subject. (Leave in terms of π.)

  3. Rearrange the formula to make the indicated variable the subject, then substitute the given values to find its value. Understanding

    1. v = u + at. Make a the subject, then find a when v = 50, u = 10 and t = 8.
    2. P = 2(l + w). Make w the subject, then find w when P = 36 and l = 11.
    3. I = Prt ÷ 100. Make P the subject, then find P when I = 180, r = 6 and t = 5.
    4. A = πr². Make r the subject, then find r (to 1 d.p.) when A = 50.27 cm² (use π = 3.14).
    5. E = ½mv². Make m the subject, then find m when E = 450 J and v = 15 m/s.

  4. Identify whether the rearrangements shown are correct or incorrect. For any that are incorrect, show the correct working. Understanding

    1. From d = st, a student claims t = d − s. Is this correct?
    2. From A = ½bh, a student claims b = 2A ÷ h. Is this correct?
    3. From y = 3x − 7, a student claims x = (y + 7) ÷ 3. Is this correct?
    4. From c² = a² + b², a student claims a = c − b. Is this correct?

  5. Real-world formula applications. Rearrange then evaluate in context. Problem Solving

    1. The formula for average speed is v = d ÷ t.
      1. Rearrange to make d the subject.
      2. A cyclist maintains an average speed of 22 km/h for 2 hours and 15 minutes. How far do they travel?
    2. The formula for simple interest is I = Prt ÷ 100.
      1. Rearrange to make P the subject.
      2. An investor wants to earn $600 interest over 4 years at a rate of 5% per annum. How much must they invest?
    3. Celsius and Fahrenheit are related by F = 9C⁄5 + 32.
      1. Rearrange to make C the subject.
      2. Normal body temperature is 98.6°F. Convert this to Celsius.
      3. Water boils at 100°C. What is this in Fahrenheit?
    4. A rectangular field has a perimeter of 220 m and an area of 2800 m². Use P = 2(l + w) to find l and w. (Hint: find both unknowns by forming and solving a pair of equations.)

  6. The kinematic formula s = ut + ½at² gives displacement s (m) with initial velocity u (m/s), acceleration a (m/s²) and time t (s). Problem Solving

    1. Rearrange to make u the subject.
    2. A car starts from rest (u = 0) and travels 80 m in 4 s. Find its acceleration using the rearranged formula.
    3. Using v = u + at, find the final velocity of the car from part (b).

  7. Rearrange each formula to make the indicated variable the subject. These involve fractions — multiply through before isolating. Problem Solving

    1. T = 2πr⁄v, make v the subject. (T = period, r = radius, v = speed.)
    2. R = V⁄I, make I the subject (Ohm’s Law).
    3. P = F⁄A, make A the subject (pressure formula).
    4. v² = u² + 2as, make s the subject.

  8. Two students rearrange the same formula differently. Determine which student (if either) is correct, or whether both forms are equivalent. Problem Solving

    1. Formula: E = mc². Make m the subject.
      Student A writes: m = E ÷ c².
      Student B writes: m = E ÷ (c × c).
      Are both correct? Explain.
    2. Formula: v² = u² + 2as. Make a the subject.
      Student A writes: a = (v² − u²) ÷ (2s).
      Student B writes: a = v² ÷ (2s) − u².
      Which is correct? Show the correct working.
    3. Formula: A = ½h(a + b) (area of trapezium). Make h the subject.
      Student A writes: h = 2A ÷ (a + b).
      Student B writes: h = 2A ÷ a + b.
      Which is correct? Explain the error in the incorrect version.

  9. Each scenario requires you to choose and rearrange the correct formula. Show all working. Problem Solving

    1. A circle has an area of 113.1 cm². Use A = πr² to find its radius (to 1 d.p.).
    2. A cylindrical tank has volume V = πr²h. The tank is 2 m tall and holds 25.13 m³. Find its radius (to 1 d.p.).
    3. The formula for compound interest is A = P(1 + r)² for a two-year investment. Rearrange to make P the subject, then find P if A = $1210 and r = 0.1 (10%).

  10. Extended investigation — create and apply. Problem Solving

    1. Write down a formula of your own that contains at least three variables. Rearrange it twice to make two different variables the subject. Show all steps.
    2. The stopping distance of a car is given by d = v² ÷ (2μg), where v is speed, μ is the friction coefficient, and g = 10 m/s².
      1. Rearrange to make v the subject.
      2. A car stops in 40 m on a road where μ = 0.5. What was its speed in m/s?
      3. Convert this speed to km/h. (Multiply m/s by 3.6.)
    3. A rectangular swimming pool has length l = 2w + 3, where w is the width in metres. The perimeter formula is P = 2(l + w).
      1. Substitute the expression for l into the perimeter formula and simplify.
      2. If P = 54 m, find w and then l.
      3. Find the area of the pool.
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